Composition

What is Composition?

You have two functions. You chain them — the output of the first becomes the input to the second.

$f: A \to B, \quad g: B \to C$

Composition: $g \circ f$

This reads: “g of f” or “g after f”

How It Works

$(g \circ f)(x) = g(f(x))$

First apply $f$, then apply $g$ to the result.

Example

$f(x) = x + 1$ $g(x) = x^2$

Find $(g \circ f)(2)$:

1. First: $f(2) = 2 + 1 = 3$
2. Then: $g(3) = 3^2 = 9$

$(g \circ f)(2) = 9$

Order Matters

$g \circ f \neq f \circ g$

Same functions, different order:

$(g \circ f)(2) = g(f(2)) = g(3) = 9$

$(f \circ g)(2) = f(g(2)) = f(4) = 5$

Different results.

Always apply the inner function first.

Domain Requirements

For $g \circ f$ to exist:

• The codomain of $f$ must match the domain of $g$
• Otherwise, outputs of $f$ can’t be inputs to $g$

$f: A \to B, \quad g: B \to C, \quad g \circ f: A \to C$

Properties

Associativity:

$(h \circ g) \circ f = h \circ (g \circ f)$

Grouping doesn’t matter when composing three functions.

Identity function:

$\text{id}_A(x) = x$

$f \circ \text{id}_A = f$ $\text{id}_B \circ f = f$

Composing with identity changes nothing.